Weighted polynomial approximation for weights with slowly varying extremal density
In this paper approximation of continuous functions \(f\) by weighted polynomials of the form \(w^nP_n\) on a closed subset \(\Sigma\) of the real line is investigated. The weight \(w\) is a continuous function on \(\Sigma\), not identically vanishing. The polynomials \(P_n\) have degree at most \(n\). Approximability is governed by the known theorem: There exists a closed set \(S\subset \Sigma\) such that a continuous function \(f\) on \(\Sigma\) is the uniform limit of a sequence \(w^nP_n\), as \(n\to \infty\) if and only if \(f\) vanishes on \(S\). With \(w\) one can associate a probability measure \(\mu_w\) as the unique minimum solution of a weighted energy integral. When \(\mu_w\) has a density \(d\mu_w\) which is continuous and positive near a point \(x_0\in\Sigma\), then \(x_0\notin S\). It is shown in the paper under review that the same is true if \(d\mu_w\) is slowly varying and bounded away from 0 near \(x_0\). This allows logarithmic singularities. As second main result the author shows that under mild additional assumptions, \(x_0\notin S\) holds even when the density \(d\mu_w\) has an isolated zero at \(x_0\) provided \(d\mu_w\) is slowly varying. These results are in contrast to the known results which ensure that polynomial zeros or polynomial singularities \(x_0\) where the density \(d\mu_w\) behaves like \(|x-x_0|^\lambda\) with \(\lambda>0\) or \(\lambda<0\) must lie in \(S\).
scientific article; zbMATH DE number 1346403